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35 pages, 1745 KiB

Open AccessArticle

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

by Dmytro Sytnyk and Barbara Wohlmuth

Mathematics 2023, 11(10), 2312; https://doi.org/10.3390/math11102312 - 16 May 2023

Cited by 2 | Viewed by 1225

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula [...] Read more.

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A, fractional order

α

and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side

f (t)

. The resulting method possesses exponential convergence for positive sectorial A, any finite t, including

t = 0

and the whole range

α \in (0, 2)

. It is suitable for a practically important case, when no knowledge of

f (t)

is available outside the considered interval

t \in [0, T]

. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates. Full article

(This article belongs to the Special Issue Numerical Methods for Fractional Differential Equations and Applications)

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15 pages, 1108 KiB

Open AccessArticle

A Comparative Study of the Fractional-Order Belousov–Zhabotinsky System

by Samir A. El-Tantawy, Rasool Shah, Albandari W. Alrowaily, Nehad Ali Shah, Jae Dong Chung and Sherif. M. E. Ismaeel

Mathematics 2023, 11(7), 1751; https://doi.org/10.3390/math11071751 - 6 Apr 2023

Cited by 10 | Viewed by 1531

Abstract

In this article, we present a modified strategy that combines the residual power series method with the Laplace transformation and a novel iterative technique for generating a series solution to the fractional nonlinear Belousov–Zhabotinsky (BZ) system. The proposed techniques use the Laurent series [...] Read more.

In this article, we present a modified strategy that combines the residual power series method with the Laplace transformation and a novel iterative technique for generating a series solution to the fractional nonlinear Belousov–Zhabotinsky (BZ) system. The proposed techniques use the Laurent series in their development. The new procedures’ advantages include the accuracy and speed in obtaining exact/approximate solutions. The suggested approach examines the fractional nonlinear BZ system that describes flow motion in a pipe. Full article

(This article belongs to the Special Issue Numerical Methods for Fractional Differential Equations and Applications)

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21 pages, 1216 KiB

Open AccessArticle

Elucidating the Effects of Ionizing Radiation on Immune Cell Populations: A Mathematical Modeling Approach with Special Emphasis on Fractional Derivatives

by Dalal Yahya Alzahrani, Fuaada Mohd Siam and Farah A. Abdullah

Mathematics 2023, 11(7), 1738; https://doi.org/10.3390/math11071738 - 5 Apr 2023

Viewed by 1503

Abstract

Despite recent advances in the mathematical modeling of biological processes and real-world situations raised in the day-to-day life phase, some phenomena such as immune cell populations remain poorly understood. The mathematical modeling of complex phenomena such as immune cell populations using nonlinear differential [...] Read more.

Despite recent advances in the mathematical modeling of biological processes and real-world situations raised in the day-to-day life phase, some phenomena such as immune cell populations remain poorly understood. The mathematical modeling of complex phenomena such as immune cell populations using nonlinear differential equations seems to be a quite promising and appropriate tool to model such complex and nonlinear phenomena. Fractional differential equations have recently gained a significant deal of attention and demonstrated their relevance in modeling real phenomena rather than their counterpart, classical (integer) derivative differential equations. We report in this paper a mathematical approach susceptible to answering some relevant questions regarding the side effects of ionizing radiation (IR) on DNA with a particular focus on double-strand breaks (DSBs), leading to the destruction of the cell population. A theoretical elucidation of the population memory was carried out within the framework of fractional differential equations (FODEs). Using FODEs, the mathematical approach presented herein ensures connections between fractional calculus and the nonlocal feature of the fractional order of immune cell populations by taking into account the memory trace and genetic qualities that are capable of integrating all previous actions and considering the system’s long-term history. An illustration of both fractional modeling, which provides an excellent framework for the description of memory and hereditary properties of immune cell populations, is elucidated. The mathematics presented in this research hold promise for modeling real-life phenomena and paves the way for obtaining accurate model parameters resulting from the mathematical modeling. Finally, the numerical simulations are conducted for the analytical approach presented herein to elucidate the effect of various parameters that govern the influence of ionizing irradiation on DNA in immune cell populations as well as the evolution of cell population dynamics, and the results are presented using plots and contrasted with previous theoretical findings. Full article

(This article belongs to the Special Issue Numerical Methods for Fractional Differential Equations and Applications)

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19 pages, 730 KiB

Open AccessArticle

Analyzing Both Fractional Porous Media and Heat Transfer Equations via Some Novel Techniques

by Wedad Albalawi, Rasool Shah, Nehad Ali Shah, Jae Dong Chung, Sherif M. E. Ismaeel and Samir A. El-Tantawy

Mathematics 2023, 11(6), 1350; https://doi.org/10.3390/math11061350 - 10 Mar 2023

Cited by 3 | Viewed by 2308

Abstract

It has been increasingly obvious in recent decades that fractional calculus (FC) plays a key role in many disciplines of applied sciences. Fractional partial differential equations (FPDEs) accurately model various natural physical phenomena and many engineering problems. For this reason, the analytical and [...] Read more.

It has been increasingly obvious in recent decades that fractional calculus (FC) plays a key role in many disciplines of applied sciences. Fractional partial differential equations (FPDEs) accurately model various natural physical phenomena and many engineering problems. For this reason, the analytical and numerical solutions to these issues are seriously considered, and different approaches and techniques have been presented to address them. In this work, the FC is applied to solve and analyze the time-fractional heat transfer equation as well as the nonlinear fractional porous media equation with cubic nonlinearity. The idea of solving these equations is based on the combination of the Yang transformation (YT), the homotopy perturbation method (HPM), and the Adomian decomposition method (ADM). These combinations give rise to two novel methodologies, known as the homotopy perturbation transform method (HPTM) and the Yang tranform decomposition method (YTDM). The obtained results show the significance of the accuracy of the suggested approaches. Solutions in various fractional orders are found and discussed. It is noted that solutions at various fractional orders lead to an integer-order solution. The application of the current methodologies to other nonlinear fractional issues in other branches of applied science is supported by their straightforward and efficient process. In addition, the proposed solution methods can help many plasma physics researchers in interpreting the theoretical and practical results. Full article

(This article belongs to the Special Issue Numerical Methods for Fractional Differential Equations and Applications)

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11 pages, 697 KiB

Open AccessArticle

A Novel Approach to Solving Fractional-Order Kolmogorov and Rosenau–Hyman Models through the q-Homotopy Analysis Transform Method

by Laila F. Seddek, Essam R. El-Zahar, Jae Dong Chung and Nehad Ali Shah

Mathematics 2023, 11(6), 1321; https://doi.org/10.3390/math11061321 - 9 Mar 2023

Cited by 2 | Viewed by 1437

Abstract

In this study, a novel method called the q-homotopy analysis transform method (q-HATM) is proposed for solving fractional-order Kolmogorov and Rosenau–Hyman models numerically. The proposed method is shown to have fast convergence and is demonstrated using test examples. The validity of the proposed [...] Read more.

In this study, a novel method called the q-homotopy analysis transform method (q-HATM) is proposed for solving fractional-order Kolmogorov and Rosenau–Hyman models numerically. The proposed method is shown to have fast convergence and is demonstrated using test examples. The validity of the proposed method is confirmed through graphical representation of the obtained results, which also highlights the ability of the method to modify the solution’s convergence zone. The q-HATM is an efficient scheme for solving nonlinear physical models with a series solution in a considerable admissible domain. The results indicate that the proposed approach is simple, effective, and applicable to a wide range of physical models. Full article

(This article belongs to the Special Issue Numerical Methods for Fractional Differential Equations and Applications)

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15 pages, 526 KiB

Open AccessArticle

An Innovative Approach to Nonlinear Fractional Shock Wave Equations Using Two Numerical Methods

by Meshari Alesemi

Mathematics 2023, 11(5), 1253; https://doi.org/10.3390/math11051253 - 5 Mar 2023

Viewed by 1209

Abstract

In this research, we propose a combined approach to solving nonlinear fractional shock wave equations using an Elzaki transform, the homotopy perturbation method, and the Adomian decomposition method. The nonlinear fractional shock wave equation is first transformed into an equivalent integral equation using [...] Read more.

In this research, we propose a combined approach to solving nonlinear fractional shock wave equations using an Elzaki transform, the homotopy perturbation method, and the Adomian decomposition method. The nonlinear fractional shock wave equation is first transformed into an equivalent integral equation using the Elzaki transform. The homotopy perturbation method and Adomian decomposition method are then utilized to approximate the solution of the integral equation. To evaluate the effectiveness of the proposed method, we conduct several numerical experiments and compare the results with existing methods. The numerical results show that the combined method provides accurate and efficient solutions for nonlinear fractional shock wave equations. Overall, this research contributes to the development of a powerful tool for solving nonlinear fractional shock wave equations, which has potential applications in many fields of science and engineering. This study presents a solution approach for nonlinear fractional shock wave equations using a combination of an Elzaki transform, the homotopy perturbation method, and the Adomian decomposition method. The Elzaki transform is utilized to transform the nonlinear fractional shock wave equation into an equivalent integral equation. The homotopy perturbation method and Adomian decomposition method are then employed to approximate the solution of the integral equation. The effectiveness of the combined method is demonstrated through several numerical examples and compared with other existing methods. The results show that the proposed method provides accurate and efficient solutions for nonlinear fractional shock wave equations. Full article

(This article belongs to the Special Issue Numerical Methods for Fractional Differential Equations and Applications)

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14 pages, 470 KiB

Open AccessArticle

Non-Overlapping Domain Decomposition via BURA Preconditioning of the Schur Complement

by Nikola Kosturski, Svetozar Margenov and Yavor Vutov

Mathematics 2022, 10(13), 2327; https://doi.org/10.3390/math10132327 - 3 Jul 2022

Cited by 2 | Viewed by 1451

Abstract

A new class of high-performance preconditioned iterative solution methods for large-scale finite element method (FEM) elliptic systems is proposed and analyzed. The non-overlapping domain decomposition (DD) naturally introduces coupling operator at the interface

γ

. In general,

γ

is a manifold of lower [...] Read more.

A new class of high-performance preconditioned iterative solution methods for large-scale finite element method (FEM) elliptic systems is proposed and analyzed. The non-overlapping domain decomposition (DD) naturally introduces coupling operator at the interface

γ

. In general,

γ

is a manifold of lower dimensions. At the operator level, a key property is that the energy norm associated with the Steklov-Poincaré operator is spectrally equivalent to the Sobolev norm of index 1/2. We define the new multiplicative non-overlapping DD preconditioner by approximating the Schur complement using the best uniform rational approximation (BURA) of

L_{γ}^{1 / 2}

. Here,

L_{γ}^{1 / 2}

denotes the discrete Laplacian over the interface

γ

. The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the BURA-based non-overlapping DD preconditioner has optimal computational complexity

O (n)

, where n is the number of unknowns (degrees of freedom) of the FEM linear system. All theoretical estimates are robust, with respect to the geometry of the interface

γ

. Results of systematic numerical experiments are given at the end to illustrate the convergence properties of the new method, as well as the choice of the involved parameters. Full article

(This article belongs to the Special Issue Numerical Methods for Fractional Differential Equations and Applications)

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18 pages, 341 KiB

Open AccessArticle

On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type

by Raimondas Čiegis and Ignas Dapšys

Mathematics 2022, 10(13), 2155; https://doi.org/10.3390/math10132155 - 21 Jun 2022

Cited by 3 | Viewed by 1943

Abstract

The main aim of this article is to propose a general framework for the theoretical analysis of discrete schemes used to solve multi-dimensional parabolic problems with fractional power elliptic operators. This analysis is split into three parts. The first part is based on [...] Read more.

The main aim of this article is to propose a general framework for the theoretical analysis of discrete schemes used to solve multi-dimensional parabolic problems with fractional power elliptic operators. This analysis is split into three parts. The first part is based on techniques well developed for the solution of nonlocal elliptic problems. The obtained discrete elliptic operators are used to formulate semi-discrete approximations. Next, the fully discrete schemes are constructed by applying the classical and robust approximations of time derivatives. The existing stability and convergence results are directly included in the new framework. In the third part, approximations of transfer operators are constructed by using uniform and the best uniform rational approximations. The stability and accuracy of the obtained local discrete schemes are investigated. The results of computational experiments are presented and analyzed. A three-dimensional test problem is solved. The rational approximations are constructed by using the BRASIL algorithm. Full article

(This article belongs to the Special Issue Numerical Methods for Fractional Differential Equations and Applications)

18 pages, 435 KiB

Open AccessArticle

Rational Approximations in Robust Preconditioning of Multiphysics Problems

by Stanislav Harizanov, Ivan Lirkov and Svetozar Margenov

Mathematics 2022, 10(5), 780; https://doi.org/10.3390/math10050780 - 28 Feb 2022

Cited by 7 | Viewed by 1859

Abstract

Multiphysics or multiscale problems naturally involve coupling at interfaces which are manifolds of lower dimensions. The block-diagonal preconditioning of the related saddle-point systems is among the most efficient approaches for numerically solving large-scale problems in this class. At the operator level, the interface [...] Read more.

Multiphysics or multiscale problems naturally involve coupling at interfaces which are manifolds of lower dimensions. The block-diagonal preconditioning of the related saddle-point systems is among the most efficient approaches for numerically solving large-scale problems in this class. At the operator level, the interface blocks of the preconditioners are fractional Laplacians. At the discrete level, we propose to replace the inverse of the fractional Laplacian with its best uniform rational approximation (BURA). The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the proposed preconditioners have optimal computational complexity

O (N)

, where N is the number of unknowns (degrees of freedom) of the coupled discrete problem. The main theoretical contribution is the condition number estimates of the BURA-based preconditioners. It is important to note that the obtained estimates are completely analogous for both positive and negative fractional powers. At the end, the analysis of the behavior of the relative condition numbers is aimed at characterizing the practical requirements for minimal BURA orders for the considered Darcy–Stokes and 3D–1D examples of coupled problems. Full article

(This article belongs to the Special Issue Numerical Methods for Fractional Differential Equations and Applications)

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